THE PROPAGATOR MATRIX RELATED TO THE KIRCHHOFF-HELMHOLTZ INTEGRAL IN INVERSE WAVE-FIELD EXTRAPOLATION

The Kirchhoff-Helmholtz formula for the wavefield inside a closed surf ace surrounding a volume is most commonly given as a surface integral over the field and its normal derivative, given the Green's function o f the problem. In this case the source point of the Green's function, or the observation point, is located inside the volume enclosed by the surface. However, when locating the observation point outside the clo sed surface, the Kirchhoff-Helmholtz formula constitutes a functional relationship between the field and its normal derivative on the surfac e, and thereby defines an integral equation for the fields. By dividin g the closed surface into two parts, one being identical to the (infin ite) data measurement surface and the other identical to the (infinite ) surface onto which we want to extrapolate the data, the solution of the Kirchhoff-Helmholtz integral equation mathematically gives exact i nverse extrapolation of the field when constructing a Green's function that generates either a null pressure field or a null normal gradient of the pressure field on the latter surface. In the case when the sur faces are plane and horizontal and the medium parameters are constant between the surfaces, analysis in the wavenumber domain shows that the Kirchhoff-Helmholtz integral equation is equivalent to the Thomson-Ha skell acoustic propagator matrix method. When the medium parameters ha ve smooth vertical gradients, the Kirchhoff-Helmholtz integral equatio n in the high-frequency approximation is equivalent to the WKBJ propag ator matrix method, which also is equivalent to the extrapolation meth od denoted by extrapolation by analytic continuation.